Abstract
Let K [k,t] be the complete graph on k vertices from which a set of edges, induced by a clique of order t, has been dropped (note that K [k,1] is just K k). In this paper we study R(K [k 1 ,t 1 ] ,. .. , K [kr,tr]) (the smallest integer n such that for any r-edge coloring of K n there always occurs a monochromatic K [k i ,t i ] for some i). We first present a general upper bound (containing the well-known Graham-Rödl upper bound for complete graphs in the particular case when t i = 1 for all i). We then focus our attention when r = 2 and dropped cliques of order 2 and 3 (edges and triangles). We give the exact value for R(K [n,2] , K [4,3]) and R(K [n,3] , K [4,3]) for all n ≥ 2.